Crude oil is mixture of hydrocarbon molecules that have from 4 carbon atoms to 50 or more. Most of the molecules in crude are long single chains of carbon atoms with hydrogen atoms attached.  Called "alkanes" or "parafins" these molecules have no double chemical bonds.

A very simple refinery would process the crude by distillation alone.  The distilling process would seperate the crude into smaller fractions of mixtures of similar hydrocarbons.  These fracions are called "straight runs" by the blender. Because they come straight from the crude without further processing. The problem is that distillation alone doesn't produce very much of the gasoline and diesel components.  A typical barrel of crude might only produce 1/2 barrel of these valuable products. The rest of the crude would be in very light components (C5 pentanes and lighter) and low value heavy fuels (C50 & greater).  

To create more gasoline and diesel components, the refiner uses further processing with heat, pressure, and exotic catalysts to break down longer chain hydrocarbons into shorter ones with the desired properties. These processes might include cat-cracking, vis-breaking, hydrocracking, coking, etc.  

During this process, some of the hydrocarbons form double bonds (a carbon atom making 2 bonds with an adjacent carbon atom).  These compounds are called "unsaturates" because they are not fully saturated by hydrogen atoms.  Double bonds are more reactive and less "stable" than single bonds. In the presence of oxygen, like in a gas or diesel can that is opened frequently, the double bonds react to form longer chain hydrocarbons, cyclic compounds and other undesired forms that can lead to gums and heavy deposits.

Adding stabilizers to the diesel or gasoline the stabilizer reacts with the double bonds to keep the heavy compounds from forming.